One way or another the duality occurs in math.

For example, let me define the dual of a pair (a, b) as (b, a) with the notation

(a, b)* := (b, a) Of course,

(a, b)** = (b, a)* = (a, b). But this duality is more touchy : for example, we can prove diff( (f, g)*) = (diff(f, g)) * and so on.

Another example of duality : Let A = { 1,2,3, 4 } be a set of individuals x and f, g, h one-to-one maps on A onto K = {a, b, c, d,}

- | f, g, h

1 | a b c

2 | b c a

3 | c d b

4 | d a d

We have f(1)= a, f(2)= b,..., h(4) = d.

There is a risk that someone that does not know the 'right' notations to see the table as functions defined on { f, g, h } :

1(f) = a, 1(g) = b,... 4(h) = d; so, x(f) = f(x).

Thus, to maintain the coherence of 'right' notations, we write x*(F) instead of x(F) to say the same thing. Of course, x**(F) = (x*(F))* = (F(x))*= F*(x) = x(F) for every F and every x.

Now comes my question : Has anyone an example where the duality brings more than a rewriting of something, when it is necessary to a result that can't be proved else ?

The first example could be the contrapositive :

**contrapositive (contrapositive (implication)) = implication**

To prove that sqrt(2) is not rational, one needs to prove the contrapositive.